4^{e}série, t. 37, 2004, p. 105 à 170.

## MAGNETIC BOTTLES FOR THE NEUMANN PROBLEM:

## CURVATURE EFFECTS IN THE CASE OF DIMENSION 3 (GENERAL CASE)

### B

Y### B

ERNARD### HELFFER

^{1}AND

### A

BDEREMANE### MORAME

ABSTRACT. – In a recent paper Lu and Pan have analyzed the asymptotic behavior, in the semi-classical regime, of the ground state energy of the Neumann realization of the Schrödinger operator in the case of dimension 3. Although these results are rather satisfactory when the magnetic field is non-constant and satisfies some generic conditions, they are not sufficient in the case of a constant magnetic field for understanding phenomena like the onset of superconductivity and more accurate results should be obtained.

In the two-dimensional case, the effects due to the curvature of the boundary were predicted by a formal analysis of Bernoff–Sternberg and finally proved by the joint efforts of Lu–Pan, Del Pino–Felmer–Sternberg and Helffer–Morame. Our aim is to analyze similar effects in dimension3. As known from physicists and roughly analyzed by Lu–Pan, it turns out that the results depend on the geometry of the boundary especially at the points where the magnetic field is tangent at the boundary. We present here the analog of the Bernoff–

Sternberg conjecture (also formulated in a different form by Pan) and prove it in the generic situation.

2004 Published by Elsevier SAS

RÉSUMÉ. – Récemment Lu et Pan ont analysé le comportement asymptotique, en régime semi-classique, de la plus petite valeur propre de la réalisation de Neumann d’un opérateur de Schrödinger magnétique dans le cas de la dimension3. Si les résultats obtenus sont satisfaisants lorsque le champ magnétique est variable et vérifie quelques conditions génériques, ils ne permettent pas, dans le cas d’un champ magnétique constant, de comprendre des phénomènes comme la localisation d’une fonction propre associée. Dans le cas de la dimension2, Bernoff et Sternberg avaient conjecturé, sur la base de constructions formelles, que c’était la courbure du bord qui allait être la clef de cette localisation. Ceci fut finalement prouvé par Lu–Pan, Del Pino–Sternberg et Helffer–Morame. Nous nous proposons ici d’analyser le même problème en dimension3.

La littérature physique indique (et les premiers résultats dans cette direction furent démontrés par Lu et Pan) que tout dépend de la géométrie de la frontière et plus précisément de la courbe (génériquement) où le champ magnétique est tangent au bord. Nous présentons ici l’analogue de la conjecture de Bernoff–

Sternberg dans le cas de la dimension3(conjecture aussi formulée sous une forme un peu différente par Pan) et en donnons la démonstration sous des hypothèses génériques.

2004 Published by Elsevier SAS

1This work has been partially supported by the European Union under the TMR Program EBRFXT 960001 and the ESF program SPECT.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

**1. Introduction**

Our study is motivated by a question in superconductivity.^{2} Let us roughly explain how the
question occurs in this context (see [10] or [22] for a review on this aspect and references therein).

Given a bounded open setΩ*⊂*R^{3}with smooth boundary, we are interested in the properties of
the minimizers of the so called Ginzburg–Landau functional

*H*^{1}(Ω;C)*×H*^{1}(Ω;R^{3})(ψ,*A)* *→ G(ψ,A)*
with:

*G(ψ,A) =*

Ω

(∇ −*iκA)ψ* ^{2}+*κ*^{2}
2

*|ψ|*^{2}*−*12
*dx*+*κ*^{2}

Ω

*|*curl*A−σ*curl*A|*^{2}*dx.*

(1.1)

Here*A*is a given applied magnetic potential in*H*^{1}(Ω;R^{3}),*σ*is a positive parameter introduced
for permitting a variation of its intensity and the parameter*κ >*0 reflects the properties of the
sample Ωwhich will be assumed to be large. AsΩis bounded, the existence of a minimizer
is rather standard. It is also easy to observe that the pair (0, σA)is a trivial critical point of
the functional*G*which is called a normal solution. It has also be shown, that this pair is, when
curl*A*:=*H* does not vanish, a global minimum for*σ* large enough. It is therefore natural to
discuss in function of*σ, if this pair is a local or a global minimizer. By looking at the Hessian of*
the functional at this point, this question is immediately related to the positivity of the operator:

*−(∇ −iκσA)*^{2}*−κ*^{2} inΩ,

where we get from the minimization of the functional a “magnetic” Neumann condition at the boundary.

It can be seen that the change of sign of the lowest eigenvalue of this operator occurs for*σ*
of the order of *κ. To get a more precise information, we are immediately let to analyze the*
groundstate energy (that is the lowest eigenvalue) of the Neumann realization of the Schrödinger
operator with magnetic potential in an open setΩinR^{3}:

*P*_{A}* ^{h}*=
3

*j=1*

*hD**x*_{j}*−A**j*(x)2

*,*
(1.2)

where*h*=_{κσ}^{1} *will be a small parameter. This is this problem which will be the main object of*
*the paper. We recall that an elementu*in*H*^{2}(Ω)satisfies the (magnetic) Neumann condition at
the boundary, if

*N(x)·*

(hD*−A)u*

(x) = 0, *∀x∈∂Ω,*
where*N(x)*is the normal at*x*to*∂Ω.*

2One can find in Chapter 4 of [30] a physical presentation of the problem we are considering. We particularly
emphasize on their Section 4.3 where they analyze (with partially heuristic arguments) the angular dependence of the
nucleation field. We will give a mathematical proof of, what they describe for example p. 87: “For type II superconductors
*(The authors mean**κ*1/*√*

2, but we treat only*κ**large) the above calculation shows that superconductivity is not*
entirely destroyed for*H**c*_{2}*< H < H**c*_{3}. A superconducting sheath remains close to the surface parallel to the applied
field. Conversely, when the field is decreased below *H**c*_{3}, a superconducting sheath appears at the surface before
superconductivity is restored in the bulk at*H*=*H**c*2. If the sample is a long cylinder with the applied field parallel to
the axis, the sheath will cover all the surface of the cylinder. If it is a sphere, the sheath will be restricted to a small zone
near the equatorial plane when*H*is close to*H**c*_{3}. When the field is decreased towards*H**c*_{2}the sheath will progressively
extend up to the poles”.

This realization will be denoted by *P*_{A,Ω}* ^{h,N}* but we will sometimes use shorter notations
when there is no ambiguity. If

*λ(h)*is the lowest eigenvalue and

*u*

*(x) is a corresponding normalized eigenstate, we would like to discuss the asymptotic of*

^{h}*λ(h)*as

*h→*0, with the hope to analyze in a future work the localization of

*u*

*. This problem was treated in the case of dimension2by Bernoff–Sternberg [3], Lu–Pan [19–21], Del Pino–Felmer–Sternberg [28] and Helffer–Morame [12] in a rather complete way. Later a recent work by Lu–Pan [23] was starting the analysis of the dimension3and this was complemented slightly in [13]. So this paper is the continuation of the program, with a particular emphasis on the constant magnetic field case.*

^{h}We start by establishing rough estimates, but sufficiently accurate for determining the effective order of the curvature effect.

THEOREM *1.1. – Let*Ω*be a bounded open set of*R^{3}*withC*^{∞}*boundary∂Ω. LetP*_{A,Ω}^{h,N}*be*
*the Neumann operator onL*^{2}(Ω)*associated to the Schrödinger operator with constant magnetic*
*field*(hD*−A)*^{2}*and let us assume that the vector magnetic fieldH*= curl(A)*is constant and of*
*intensity equal tob*=*|H|.*

*If*

*λ(h) = inf Sp(P*_{A,Ω}* ^{h,N}*)

*is the first eigenvalue ofP*_{A,Ω}^{h,N}*, then there exists a constantC*0*such that*
*λ(h)−bΘ*0*hC*0*h*^{4/3}*,*

(1.3)

*where*Θ0*∈*]0,1[*is an universal constant, which will be defined in (2.4).*

We observe that in the case of dimension2, the corresponding error was in*O*(h^{3/2}).

We then continue with the analysis of the curvature effects, which are the analog in dimension 3 of the results first conjectured by Bernoff–Sternberg (see [3]) in the case of dimension2. Before establishing the corresponding conjecture in the case of dimension3, let us describe the main ingredients appearing in the assumptions.

It has been observed by [23] and this will be recalled in Section 4, that the ground state*u** ^{h}*is
localized (as

*h→*0) near the boundary

*∂Ω*but more precisely on the set:

Γ*H*= *x∈∂Ω|*

*H|N*(x)

= 0
*,*
(1.4)

that is the set of points in*∂Ω*where*H* is tangent.

It is natural to assume that:

Γ*H*is a regular submanifold of*∂Ω,*
(1.5)

that is a disjoint union of regular curves. From now on, we choose an orientation on each
curve. At each point*x*ofΓ*H*, we will associate the normal curvature along the magnetic field
*H*=*H*(B)by:

*κ**n,B*(x) :=*K**x*

*T*(x)*∧N(x),* *H*

*|H|*

*,*
(1.6)

where*K*denotes the second fundamental form on the surface*∂Ω*(see Section 8) and*T*(x)is
the unit oriented tangent vector toΓ*H* at*x. It is natural to assume that:*

*κ**n,B**= 0,* onΓ*H**.*
(1.7)

The last generic assumption is:

The set of points where*H*is tangent toΓ*H*is isolated.

(1.8)

The following function onΓ*H* will play an important role:

˜
*γ*0(x) =

1 2

2/3

ˆ

*ν*0*δ*_{0}^{1/3}*κ**n,B*(x)^{2/3}

*δ*0+ (1*−δ*0)

*T*(x)
*H*

*|H|*

^{2}1/3
*,*
(1.9)

where*ν*ˆ0*>*0 and*δ*0*∈*]0,1[ are universal constants attached to spectral invariants related to
two model Hamiltonians respectively defined onRandR^{+}and which will be defined in (2.15)
and (2.7).

Associated to this function, which will play the role of effective curvature, we define:

ˆ
*γ*0= inf

*x**∈*Γ*H*

˜
*γ*0(x).

(1.10)

Our main theorem is:

THEOREM *1.2. – LetP*_{A,Ω}^{h,N}*be the Neumann realization onL*^{2}(Ω)*of the magnetic Laplacian*
(hD*−A)*^{2}*, whereh∈*]0,1[*is a small parameter andAcorresponds to constant magnetic field.*

*Under assumptions (1.5), (1.7) and (1.8), there existη >*0*andγ*ˆ0*>*0*such that:*

inf Sp(P_{A,Ω}* ^{h,N}*) =

*bΘ*0

*h*+ ˆ

*γ*0

*b*

^{2/3}

*h*

^{4/3}+

*O*(h

^{4}

^{3}

^{+η}), (1.11)

*whereγ*ˆ0*is defined in (1.10).*

We conjectured this theorem in September 2001, simultaneously with Pan [26] (who proposed
another equivalent^{3} formulation and obtained the upper bound). A first complete proof was given
in mp_arc [14] under additional non generic conditions. This paper, which is a slightly modified
version of the preprint [15], gives now a complete proof in the general generic case. Although
the methods of proof can also lead to localization results for the ground state (see [12,13]) or
more generally for minimizers of the Ginzburg–Landau functional (see [19–23,17]), this will not
be discussed here. This was actually explored in [26], under the assumption that the conjecture
was true.

The paper is organized as follows.

In Sections 2 and 3, we recall previous results extracted from [6,12,23,13], which will play
an important role in the analysis. In Section 4, we recall the results of [23] devoted to the case
when the magnetic field is not constant. It is also shown there that the problem is reduced in the
constant magnetic case to a neighborhood of the boundary. In Section 5, we make explicit our
choice of coordinates near the boundary. Section 6 gives rough upper bounds for the ground state
energy by constructing quasimodes. In Section 7, we present our first lower bounds which are
sufficiently accurate for giving the right order for the remainder. In Section 8, we go further in
the choice of adapted coordinates, taking in particular account of the fact that the magnetic field
is constant. This permits to introduce our magnetic invariants attached toΓ*H*and to present our
main results in a more precise form. Section 9 is devoted to the research of simpler models for
the magnetic potentials obtained in the adapted coordinates by suitable gauge transformations
and by neglecting “small” terms. Section 10 is devoted to the estimate of the errors done in the

3See the appendix in [15].

use of the previous models. In Section 11, we start by some heuristics which lead to the spectral analysis of a simplified model which will be used in the general case. Section 12 is devoted to the proof of the accurate upper bounds. Section 13 gives complementary lower bounds for the model. Section 14 presents the structure of the proof of accurate lower bounds according to different zones and treats the easy zones. Sections 15 and 16 correspond to the treatment of the difficult zones.

**2. The case of dimension**2
**2.1. Main results for the models with constant magnetic fields**

We first consider the operator:

*P*_{A}* ^{h}*:= (hD

*x*1

*−A*1)

^{2}+ (hD

*x*2

*−A*2)

^{2}

*,*with

*A*=

*−b*
2*x*2*,b*

2*x*1

*,* *h >*0 and *b >*0,

and analyze the spectrum of its realization inR^{2}or of its Neumann realization inR^{2}+.

We observe that by homogeneity, one can reduce the analysis to*h*= 1and*b*= 1. It is well
known that in the case of R^{2} the spectrum is a point spectrum and that the eigenvalues are
given by(2n+ 1)with (n*∈*N), each eigenvalue being of infinite multiplicity. One way, to see
this is to show the unitary equivalence (via a gauge transformation, a partial Fourier transform
and a translation) with the harmonic oscillator (D_{t}^{2}+*t*^{2})but seen as an unbounded operator
on*L*^{2}(R^{2}*t,s*).

The case of the Neumann realization inR^{2}+is a little more delicate. By unitary transformation
we get the Neumann realization of the operator of

*Q(t, D**t*;*s) :=D*_{t}^{2}+ (t*−s)*^{2} on*L*^{2}(R+*×*R),
(2.1)

which is reduced to the analysis of the family (s*∈*R) of operators:

*Q(s) :=D*^{2}* _{t}*+ (t

*−s)*

^{2}

*,*(2.2)

defined on*L*^{2}(R+).

Let*µ(s)*be defined by:

*µ(s)*is the lowest eigenvalue of the Neumann realization of*Q(s)*inR^{+}*.*
(2.3)

One easily shows (cf. [29])

PROPOSITION *2.1. – The infimum of the spectrum of the Neumann realization ofP*_{A}^{h}*is given*
*bybhΘ*0*with*

Θ0:= inf

*s**∈R**µ(s).*

(2.4)

**2.2. An important one-dimensional family of operators on**R

We recall the main properties of the groundstate energy of the Neumann realization onR^{+}of
*Q(s) :=D*^{2}* _{t}*+ (t

*−s)*

^{2}

*,*

(2.5)

in function of the parameter *s∈*R. Let us just list some properties of *µ(s)* and of the
corresponding normalized eigenvector*ϕ** ^{s}*and refer to [6,12] for proofs or details. The Neumann
eigenvalue

*µ(s)*satisfies the following properties:

1.lim*s**→−∞**µ(s) = +∞*;
2.*µ*is decreasing for*s <*0;

3.*µ(0) = 1;*

4.lim*s**→*+*∞**µ(s) = 1;*

5.*µ*admits in]0,+∞[a unique minimumΘ0*<*1at some*ξ*0*>*0. So
Θ0:= inf

*s**∈R**µ(s) =µ(ξ*0)*<*1.

(2.6)

This minimum is nondegenerate and more precisely:

0*< δ*0:=1

2*µ** ^{}*(ξ0)

*<*1.

(2.7)

6. We introduce*ϕ*0=*ϕ*^{ξ}^{0}. We have:

R+

(t*−ξ*0)*ϕ*0(t)^{2}*dt*= 0,
(2.8)

which just corresponds to the condition*µ** ^{}*(ξ0) = 0.

7.*ϕ*0is rapidly decreasing at*∞*.

Let us just mention that the proof of some of these properties is based on the following identity,
relating the first eigenfunction*ϕ** ^{s}*and

*µ(s):*

*ϕ*^{s}^{2}*µ** ^{}*(s) =

*s*^{2}*−µ(s)*
*ϕ** ^{s}*(0)2

*.*
(2.9)

**2.3. Applications: main results in dimension**2

As an application of the analysis of the models, one gets (cf. [21]) via a partition of unity for the lower bound and a suitable construction of quasimodes for the upper bound, the following general result (in the two-dimensional case):

THEOREM *2.2. – Ifλ(h)* *is the lowest eigenvalue of the Neumann realization ofP*_{A}^{h}*in*Ω,
*then we have:*

*h*lim*→*0

*λ(h)*

*h* = min

*x*inf*∈*Ω

*B(x),*Θ0 inf

*x**∈**∂Ω*

*B*(x)*,*
(2.10)

*whereB*= curl*A.*

We emphasize that there is no assumption that the magnetic field is constant. In the case of the
constant magnetic field, one can actually have more precise results for*λ(h). WhenB*is constant,
the minimum in (2.10) is obtained by the second term and we showed in [3,28,12] the

THEOREM *2.3. – If the magnetic field is constant of intensityb, then:*

*λ(h) =bΘ*0*h−c*1*b*^{1/2}
sup

*x**∈**∂Ω*

*κ(x)*

*h*^{3/2}+ o(h^{3/2}),
(2.11)

*wherec*1*is a universal strictly positive universal constant and* *κ(x)is the curvature of∂Ωat*
*x∈∂Ω.*

**2.4. The Montgomery’s model and a second important family of operators on**R

We just discuss a model that we shall meet indirectly later and which is interesting. It first
appears in [24] but see also [11]. We consider inR^{2}*x,y*, and for some parameter*κ >*0the operator:

*P*:=*h*^{2}*D*^{2}* _{x}*+

*hD**y**−κ*
2*x*^{2}

2

*.*
(2.12)

The magnetic potential is*A&*= (0,^{κ}_{2}*x*^{2})and we have:

curl*A*=*κx.*

So the magnetic field vanishes along the line *{x*= 0}. Let us briefly describe the spectral
analysis. After a Fourier transform in the*y-variable, we first get:*

*P*ˆ=*h*^{2}*D*_{x}^{2}+

*hη−κ*
2*x*^{2}

2

*,*

and this leads to the analysis of the family, parametrized by *η∈*R, of selfadjoint operators
on*L*^{2}(R):

*P*ˆ(η) =*h*^{2}*D*^{2}* _{x}*+

*hη−κ*
2*x*^{2}

2

*.*
Using a simple dilation, we get:

inf Sp( ˆ*P*) = inf

*η* inf Sp*P*ˆ(η)

=*h*^{4/3}
*κ*

2
^{2/3}inf

*ρ* inf Sp

*D*^{2}* _{r}*+ (r

^{2}

*−ρ)*

^{2}

*.*(2.13)

Let us recall some properties of the family of operators
*S(ρ) =D*^{2}*r*+ (r^{2}*−ρ)*^{2}*,*
(2.14)

and of the corresponding ground state*ψ** ^{ρ}*, which were established in [24,11] and [27].

1. There exists a unique*ρ*=*ρ*minsuch that:

ˆ
*ν*0:= inf

*ρ* inf Sp

*D*_{r}^{2}+ (r^{2}*−ρ)*^{2}

= inf Sp

*D*^{2}* _{r}*+ (r

^{2}

*−ρ*min)

^{2}

*.*(2.15)

2.*ψ** ^{ρ}*belongs to

*S(R)*and is even.

We shall later use the notation:

*ψ*0=*ψ*^{ρ}^{min}*.*
(2.16)

**3. Constant magnetic field: models in**R^{3}**and in**R^{3}+

As in the case of dimension2, where the first thing was to understand the model with constant
magnetic field inR^{2}andR^{2}+, we shall discuss the case ofR^{3}andR^{3}+.

**3.1. Model in**R^{3}

We start (after some gauge transformation) of the Schrödinger operator with constant magnetic field in dimension3.

*P** ^{h}*(

*H) :=&*

*h*

^{2}

*D*

^{2}

_{x}_{1}+ (hD

*x*

_{2}

*−H*3

*x*1)

^{2}+ (hD

*x*

_{3}+

*H*2

*x*1

*−H*1

*x*2)

^{2}

*.*(3.1)

The following lemma has the age of quantum mechanics:

LEMMA *3.1. – The bottom of the spectrum of the selfadjoint realization ofP** ^{h}*(

*H&*)

*in*R

^{3}

*is*inf Sp

*P*_{R}* ^{h,N}*3 (

*H&*)

=*bh,*
(3.2)

*where*

*b*=*|H|*=

*H*_{1}^{2}+*H*_{2}^{2}+*H*_{3}^{2}
*is the intensity ofH.*

**3.2. Models in halfspaces**

We refer to [23] and [13] for the proof of the results presented in this subsection. If*N*is a unit
vector inR^{3}, we now consider the Neumann realization inΩ :=*{x∈*R^{3}*|x·N >*0}. After a
rotation, we can assume in the proofs that*N*= (1,0,0), soΩisR^{3}+:=*{x*1*>*0}.

After scaling, we can assume that*h*= 1and*|H|*= 1.

After some rotation in the (x2*, x*3)variables, we can assume that the new magnetic field is
(β1*, β*2*,*0)and we are reduced to the problem of analyzing:

*P*(β1*, β*2) :=*D*_{x}^{2}_{1}+*D*^{2}_{x}_{2}+ (D*x*_{3}+*β*2*x*1*−β*1*x*2)^{2}*,*
in*{x*1*>*0}, where:

*β*_{1}^{2}+*β*^{2}_{2}= 1.

We introduce:

*β*2= cos*ϑ,* *β*1= sin*ϑ,*
and we observe that, if*N*is the external normal to*x*1= 0, we have:

*H&* *|N*=*−*sin*ϑ.*

(3.3)

By partial Fourier transform, we arrive to:

*L(ϑ, τ*) =*D*^{2}_{x}_{1}+*D*^{2}_{x}_{2}+ (τ+ cosϑx1*−*sin*ϑx*2)^{2}*,*
(3.4)

in*x*1*>*0 and with Neumann condition on*x*1= 0. The bottom of the spectrum of *L(ϑ, τ)*is
given by:

*σ(ϑ) := inf Sp*

*L(ϑ, D**t*)

= inf

*τ*

inf Sp

*L(ϑ, τ)*
*.*
(3.5)

PROPOSITION *3.2. – The bottom of the spectrum of the Neumann realization ofP** ^{h}*(

*H&*)

*in*Ω :=

*{x∈*R

^{3}

*|x·N >*0}

*is:*

inf Sp*P*_{Ω}* ^{h,N}*(

*H&*) =

*σ(ϑ)bh,*(3.6)

*whereϑ∈*[−^{π}_{2}*,*^{π}_{2}]*is defined by (3.3).*

By symmetry considerations, we observe also that:

*σ(ϑ) =σ(−ϑ) =σ(π−ϑ).*

(3.7)

It is consequently enough to look at the restriction to[0,^{π}_{2}].

**3.3. Properties of***ϑ→σ(ϑ)*

Let us now list the main properties of the function *ϑ→σ(ϑ)* on [0,^{π}_{2}]. Most of them are
established in [23] but see also [13].

1.*σ*is continuous on[0,^{π}_{2}].

2.

*σ(0) = Θ*0*<*1.

(3.8) 3.

*σ*
*π*

2

= 1.

(3.9) 4.

*σ(ϑ)*Θ0(cos*ϑ)*^{2}+ (sin*ϑ)*^{2}*.*
(3.10)

5. If *ϑ∈*]0,^{π}_{2}[, the spectrum of*L(ϑ, τ)* is independent of *τ* and its essential spectrum is
contained in[1,+∞[.

6. For*ϑ∈*]0,^{π}_{2}[,*σ(ϑ)*is an isolated eigenvalue of*L(ϑ, τ*), with multiplicity one.

7. The function*σ*is strictly increasing on[0,^{π}_{2}[.

8. The function*σ*has the following expansion for*ϑ*small:

*σ(ϑ)∼*Θ0+

*n*1

*α**n**|ϑ|*^{n}*,*
(3.11)

with*α*1=*√*
*δ*0=

*µ** ^{}*(ξ0)/2

*>*0.

A first consequence of this analysis is

PROPOSITION *3.3. – When* *b*=*|H|* *is fixed the bottom of the spectrum of* *P*_{Ω}* ^{h,N}*(

*H&*)

*in*Ω :=

*{x·N >*0

*}is minimal whenϑ*= 0

*that is, according to (3.3), that is when the magnetic*

*field vector satisfiesH&*

*·N*= 0.

**4. First results for general magnetic fields**

In the case of dimension2, under the assumption thatcurl*A*:=*B(x)>*0, the basic estimate
at the interior was the inequality:

*h*

*B(x)u(x)*^{2}*dx* (h*∇ −iA)u*^{2}*dx,* *∀u∈C*_{0}* ^{∞}*(Ω).

(4.1)

This was not enough for understanding the Neumann problem. One should more carefully
analyze the case ofR^{2}+ and, in the case of the constant magnetic field, one should also analyze
more complicate models (like for example the case of the disk). The most spectacular result was:

THEOREM *4.1. – When the magnetic field is constant, the groundstate of the Neumann*
*realization ofP*_{A}^{h}*in an open regular bounded set*Ω*⊂*R^{2} *is localized in the neighborhood of*
*the points of the boundary of maximal curvature.*

In the case of dimension 3, estimate (4.1) should be replaced by the weaker estimate established in [11] (Theorem 3.1)

THEOREM *4.2. – There existCandh*0*>*0*such that, for allh∈*]0, h0], we have:

*h*

Ω

*H(x)−Ch*^{1/4}*u(x)*^{2}*dx*

Ω

(h∇ −*iA)u*^{2}*dx,* *∀u∈C*0* ^{∞}*(Ω).

(4.2)

If this result is essentially sufficient for analyzing the Dirichlet problem inΩ, it is necessary to
implement the analysis given in the first part in order to treat the Neumann problem. Near each
point of the boundary*x, we have to use the lower bound obtained for the model with constant*
magnetic field*H*=*H*(x). Following for example the proof in [12] and using the same partition
of unity, we get:

THEOREM 4.3. –
*h*

Ω

*W**h*(x)*u(x)*^{2}*dx* (h∇ −*iA)u*^{2}*dx,* *∀u∈H*^{1}(Ω),
(4.3)

*where*

*W**h*(x) =

*|H*(x)*| −Ch*^{1/4}*,* *if* *d(x, ∂Ω)*2h^{3/8}*,*

*|H*(s(x))|σ(ϑ(x))*−Ch*^{1/4} *if* *d(x, ∂Ω)*2h^{3/8}*.*
(4.4)

Here we recall that*ϑ(x)*satisfies:

*H*

*s(x)·*sin*ϑ(x) =−*
*H*

*s(x)*

*|N*
*s(x)*

*,*
(4.5)

where*s(x)*is, for*x*near*∂Ω, the point in∂Ω*such that:

*d(x, ∂Ω) =d*
*x, s(x)*

*,*
and we observe that, due to (3.7),*σ(ϑ(x))*is well defined by (4.5).

The first consequence (compare with Theorem 2.2) is:

THEOREM 4.4. –

*h*lim*→*0

*λ(h)/h*

= min

*x*inf*∈*Ω

*H*(x)*,* inf

*x**∈**∂Ω*

*H*(x)*σ*
*ϑ(x)*

*.*
(4.6)

The lower bound is a direct consequence of Theorem 4.3 and the proof of the upper bound is
sketched in [23]. When*H*(x)is constant of intensity*b, then the minimum in (4.6) is obtained*
for the second term and we have:

*h*lim*→*0

*λ(h)/h*

=*b*

*x*inf*∈**∂Ω**σ*
*ϑ(x)*

*.*
(4.7)

In view of (6.3), there exists a point*x*such that*ϑ(x) = 0. So we get:*

*h*lim*→*0

*λ(h)/h*

= Θ0*b.*

(4.8)

*Remark 4.5. – As in [12], this leads to localization theorems ([23,13], see also [28]). In*
particular, if the magnetic field is constant, then the ground state is localized near the points
of the boundary where the magnetic field is parallel to the tangent space.

**Application.** An interesting case is the case when*H*is constant and whenΩis the ellipsoid:

*a*1*x*^{2}1+*a*2*x*^{2}2+*a*3*x*^{2}31.

The set of points where*H&* = (H1*, H*2*, H*3)is parallel is obtained by intersecting the boundary
of the ellipsoid with the plane:

*a*1*x*1*H*1+*a*2*x*2*H*2+*a*3*x*3*H*3= 0.

More generally, if the surface is strictly convex and if the magnetic field is constant, it is
possible to show that the set of points of the boundary where*H* is parallel to the tangent space
is a*C** ^{∞}*curve.

We emphasize that Theorem 4.4 does not explain all the situation. In the case with constant magnetic field it would be nice to show the role of some curvature in the localization as in the case of dimension2. This is actually our goal to give an answer to this question.

**5. Adapted coordinates**
**5.1. Magnetic geometrical invariants**

The standard coordinates on R^{3} will be denoted by *x*= (x1*, x*2*, x*3) and the standard flat
metric by*g*0. We will also use*.|.*for*g*0(. , .)or for*g*^{}_{0}(. , .), and*|X|*for(X*|X)*^{1/2}, (if*X*is
a vector field or a one-form). The standard volume onR^{3}will be denoted by

*ω*3=*dx*^{3}=*dx*1*∧dx*2*∧dx*3

and will fix also the orientation ofR^{3}.

Let*A*be a smooth magnetic potential one-form onΩ:

*A*=
3
*j=1*

*A**j*(x)*dx**j**,* *A**j**∈C** ^{∞}*( Ω;R).

The magnetic field*B*is the two-form:

*B*=*dA*=

1*i<j*3

*B**ij*(x)*dx**i**∧dx**j**,* *B**ij*(x) =*∂A**j*

*∂x**i*

(x)*−∂A**i*

*∂x**j*

(x).

(5.1)

If needed*B**ij*(x)is extended as an antisymmetric matrix*B**x*. The intensity of the magnetic field
is the non-negative function

*|B|(x) = sup* *|B**x*(X, Y)|; *X, Y* *∈T**x*Ω, *|X|*=*|Y|*= 1

*,* *∀x∈*Ω.

(5.2)

One associates to the magnetic field*B, a vector fieldH&* :=*H(B)∈T*Ωby
*B**x*(X, Y) =*ω*3

*X, Y, H**x*(B)

*,* *∀X, Y* *∈T**x*Ω.

(5.3)

In the initial coordinates, we have:

*H*(B) =
3

*j=1*

*b**j*(x) *∂*

*∂x**j*

*,* with*b*1=*B*23*, b*2=*B*31and*b*3=*B*12*.*
(5.4)

Then the magnetic intensity satisfies:

*|B|*=*H(B).*
(5.5)

We associate to the magnetic potential*A*the sesquilinear form on*H*^{1}(Ω):

*u→q*_{A}* ^{h}*(u) =

Ω

*√*

*−1h du*+*uA*^{2}*dx*^{3}*.*
(5.6)

Here*h∈*]0,+∞[is a parameter. The associated differential operator is the magnetic Laplacian
which is given in the standard coordinates by

*P*_{A}* ^{h}*= (hD

*−A)*

^{2}= 3

*j=1*

*hD**x*_{j}*−A**j*(x)2

*.*
(5.7)

**5.2. Local coordinates**

If*y*= (y1*, y*2*, y*3)are new local coordinates, that is, if *V**x*0 is a neighborhood of some point
*x*0*∈*Ωand if*x→y*= Φ(x)is a diffeomorphism of*V**x*0 onto*O ⊂*R^{3}, then in the new basis
(_{∂y}^{∂}

1*,*_{∂y}^{∂}

2*,*_{∂y}^{∂}

3)of*TV**x*_{0}, the standard metric*g*0and the volume*ω*3become
*g*0=

1*i,j*3

*g**ij**dy**i**⊗dy**j**,*
(5.8)

and

*ω*3=*|g|*^{1/2}*dy*1*∧dy*2*∧dy*3*,*
(5.9)

with

*g**ij*=
*∂x*

*∂y**i*

*∂x*

*∂y**j*

*,* *X|Y*=

1*i,j*3

*g**ij**X**i**Y**j**,* *|g|*= de t(g*ij*),
where

*X*=

*j*

*X**j*

*∂*

*∂y**j*

and *Y* =

*j*

*Y**j*

*∂*

*∂y**j*

*.*
The magnetic potential is given in the new coordinates by

*A*=
3
*j=1*

*A**j**dy**j* with*A**j*=
3
*k=1*

*A**k*

*∂x**k*

*∂y**j*

*.*
(5.10)

The magnetic field is also given by

*B*=

1*i<j*3

*B**ij**dy**i**∧dy**j* with*B**ij*=*∂A**j*

*∂y**i* *−∂A**i*

*∂y**j*

*.*
(5.11)

So the vector magnetic field becomes

*H*(B) =
3

*j=1*

˜*b**j*

*∂*

*∂y**j*

*,*
(5.12)

with

˜*b*1=*|g|*^{−}^{1/2}*B*23*,* ˜*b*2=*|g|*^{−}^{1/2}*B*31*,* ˜*b*3=*|g|*^{−}^{1/2}*B*12*,*
(5.13)

and the intensity of the magnetic field becomes:

*|B|*=

*i,j*

*g**ij*˜*b**i*˜*b**j*

1/2

*.*
(5.14)

The sesquilinear form takes the following form, for*u*supported in*V**x*0,
*q*_{A}* ^{h}*(u) =

*V**x*0

*|g|*^{1/2}

1*i,j*3

*g** ^{ij}*[hD

*y*

_{i}*u−A*

*i*

*u]×*[hD

*y*

_{j}*u−A*

*j*

*u]dy*

^{3}

*,*(5.15)

and the associated differential operator is
*P*_{A}* ^{h}*=

*|g|*

^{−}^{1/2}

1*i,j*3

(hD*y**j* *−A**j*)*· |g|*^{1/2}*g** ^{ij}*(hD

*y*

*i*

*−A*

*i*).

(5.16)

Here*g** ^{ij}*is the inverse matrix of the matrix

*g*

*ij*.

**5.3. Adapted local coordinates near the boundary**

Let *φ(x) = (y*1*, y*2)be local coordinates on the boundary and*G* the induced metric by*g*0

on *∂Ω* in these coordinates. Then for *ε >*0 small enough (and modifying a little *V**x*0 if
necessary), we can define local coordinates on*V**x*0,

Φ :*V**x*_{0}*→ S ×*]0, ε[, Φ(x) = (y1*, y*2*, y*3),
where*S*is an open set inR^{2}, such that

*y*3(x) = dist

*x, φ*^{−}^{1}(y1*, y*2)

= dist(x, ∂Ω);

(5.17) so

*x*=*φ*^{−}^{1}(y1*, y*2) +*y*3*N*

*φ*^{−}^{1}(y1*, y*2)

*,* *∀x∈ V**x*0*,*
(5.18)

where*N(x)*is the interior unit normal to*∂Ω*at the point*x∈∂Ω.*

Then we get by simple computation the form of the standard flat metric *g*0 in these new
coordinates,

*g*0=

1*i,j*3

*g**ij**dy**i**⊗dy**j*

=*dy*3*⊗dy*3+*G*+ 2y3

1*i,j*2

*∂N*

*∂y**i*

*∂x*

*∂y**j*

*dy**i**⊗dy**j*

+*y*_{3}^{2}

1*i,j*2

*∂N*

*∂y**i*

*∂N*

*∂y**j*

*dy**i**⊗dy**j**.*
(5.19)

In this case we have, using (5.10) and (5.18),
*A*3=

3
*j=1*

*A**j**N**j**.*

*Remark 5.1. – In the following, we choose a finite family of open setsV**x** _{j}*(x

*j*

*∈∂Ω) covering*a tubular neighborhood of

*∂Ω. When we speak later of local coordinates, we mean that we take*some element belonging to this family.

**6. Rough upper bounds**

To prove our results, we need some rough preliminary estimates (with some control of the remainder) of the ground state energy.

PROPOSITION *6.1. – Let* Ω*be a bounded open set of* R^{3} *with smooth boundary* *∂Ω. Let*
*P*_{A,Ω}^{h,N}*be the Neumann operator onL*^{2}(Ω)*associated to the Schrödinger operator with constant*
*magnetic field* (hD*−A)*^{2} *and let us assume that the vector magnetic field* *H* = curl(A) *is*
*constant of intensityb.*

*Ifλ(h) = inf Sp(P*_{A,Ω}* ^{h,N}*)

*is the first eigenvalue ofP*

_{A,Ω}

^{h,N}*, then there exists a constantC*0

*such*

*that*

*λ(h)bΘ*0*h*+*C*0*h*^{4/3}*.*
(6.1)

*Remark 6.2. – It is not necessary to assume that the magnetic field is constant as the proof of*
this proposition will show. If one choose a point*x*0such that*H(x*0)is tangent to*∂Ω, one can*
get the same result with *b*=*|H*(x0)|. One can then optimize by choosing a point of this kind
such that*|H*(x0)|is minimal. IfΓ*H*is the set introduced in (1.4) we will prove:

*λ(h)*Θ0 inf

*x**∈*Γ*H*

*H*(x)*h*+*C*0*h*^{4/3}*.*
(6.2)

This kind of estimate, with a more explicit, but not optimal*C*0, appears already in the appendix
of [23].

*Proof of Proposition 6.1. – As the flux of the magnetic field through the boundary is zero,*

*∂Ω*

curl
*A(x)*

*|N*(x)
*ds*=

Ω

div curl(A)*dx*= 0,
(6.3)

if*N* is the interior normal unit of*∂Ω, there existsx*0*∈∂Ω*such that the vector magnetic field is
tangent to*∂Ω*at*x*0:

curl
*A(x*0)

*|N(x*0)

= 0.

We take local coordinates(y1*, y*2)in a neighbourhood of*W*0of*x*0in*∂Ω*such that the metric is
*δ**ij*at*x*0and_{∂y}^{∂}

1 is parallel to the vector magnetic field.

We consider the adapted coordinates *y* = (y1*, y*2*, y*3), with *y*3(x) =*d(x, ∂Ω),* and then
*g**ij*(x0) =*δ**ij*. If*A*is the magnetic potential in the new coordinates,*A·dx*=*A·dy, then*

*∂A*3

*∂y*2

(x0)*−∂A*2

*∂y*3

(x0) =*b,*

*∂A*1

*∂y*3

(x0)*−∂A*3

*∂y*1

(x0) = 0, (6.4)

*∂A*2

*∂y*1

(x0)*−∂A*1

*∂y*2

(x0)) = 0.

We can find a gauge transformexp*i*^{ϕ}* _{h}*, with

*ϕ*real, such that, in (8.27),

*A*1(x0) = 0,

*A*2(x0) = 0,

*A*3= 0.

We first estimate the error occuring when eliminating the terms vanishing to order3.

If*u∈H*^{1}(Ω)is such that

supp(u)*⊂* *x;* *|x−x*0*|h** ^{δ}*
(6.5)

with*δ >*0, and if*h*is small enough, then, for some*C >*0,
*q*_{A}* ^{h}*(e

^{i}

^{ϕ}

^{h}*u)*(1 +

*h*

^{δ}*C)q*

^{h}*0(u) +*

_{A}*C*

*h*^{3δ}

*q*_{A}* ^{h}*0(u)1/2

*· u*+*h*^{6δ}*u*^{2}
*,*
(6.6)

where

*A*^{0}_{1}=*R*1(y), *A*^{0}_{2}=*−by*3+*R*2(y), *A*^{0}_{3}= 0,
the*R**j*(y)are polynomial functions, homogeneous of order two,

*R**j*(y) =

*|**α**|*=2

*a**j,α**y*^{α}*,*

and

*q*^{h}* _{A}*0(u) =

*|**y**|**Ch*^{δ}*, y*_{3}*>0*

(hD*y**u−A*^{0}(y)u^{2}*dy.*

(6.7)

We take the function*u*in the form

*u(y*1*, y*2*, y*3) :=*e*^{−}^{ib}^{1/2}^{y}^{2}^{ξ}^{0}^{/h}^{1/2}*v(y)*
with

*v(y*1*, y*2*, y*3) =*h*^{−}^{1}^{4}^{−}^{δ}*ϕ*0(b^{1/2}*h*^{−}^{1/2}*y*3)χ(4h^{−}^{δ}*y*3)χ

4h^{−}* ^{δ}*(y1

^{2}+

*y*2

^{2})

^{1/2}

*.*(6.8)

Here*χ*is a cut off function equal to one on[−^{1}_{2}*,*^{1}_{2}]and supported in[−1,1],*ϕ*0is introduced in
Section 2.2 and*δ∈*]0,^{1}_{2}[.

Observing that:

R+

*D**t**ϕ*0(t)^{2}+(t*−ξ*0)ϕ0(t)^{2}

*dt*= Θ0

R+

*ϕ*0(t)^{2}*dt,*

and that*t→ϕ*0(t)is an exponentially decreasing function at infinity, we get when*δ*=^{1}_{3}*,*
*q*^{h}*A*ˆ^{0}(u)(bΘ0*h*+*Ch*^{4/3})u^{2}*,*

(6.9) where

*A*ˆ^{0}1= 0, *A*ˆ^{0}2=*−by*3*,* *A*ˆ^{0}3= 0.

We then have to compare*q*^{h}_{A}_{ˆ}_{0}(u)and*q*^{h}* _{A}*0(u). The first try could be to use:

*q*^{h}* _{A}*0(e

^{i}

^{ϕ}

^{h}*u)*(1 +

*h*

^{δ}*C)q*

^{h}

_{A}_{ˆ}0(u) +

*C*

*h*

^{2δ}

*q*^{h}_{A}_{ˆ}0(u)1/2

*· u*+*h*^{4δ}*u*^{2}
*.*
(6.10)

This leads to error terms of size*h*^{1+δ},*h*^{1}^{2}^{+2δ},*h*^{4δ}and*h*^{2}^{−}^{2δ}. But this leads only, using (6.9) and
(6.10) with*δ*=^{1}_{3}, to (6.1) with*O(h*^{7/6})instead of*O(h*^{4/3})as expected.

In order to get effectively (6.1), we need to use (2.8). We have indeed to analyze more carefully
the term which was bounded from above in (6.10) by*h*^{2δ}(q^{h}_{A}_{ˆ}_{0}(u))^{1/2}*· u*.

The terms which were estimated by*h*^{1}^{2}^{+2δ} are of the form
*h*^{−}^{1}^{2}^{−}^{2δ}

*y**j**y**k*

*ξ*0*−b*^{1/2}*y*3

*h*^{1/2}

*ϕ*^{2}_{0}(b^{1/2}*h*^{−}^{1/2}*y*3)χ^{2}

4h^{−}* ^{δ}*(y

^{2}

_{1}+

*y*

^{2}

_{2})

^{1/2}

*dy,*with

*j*and

*k*equal to1or2. But they actually vanish due to (2.8).

The other terms have actually better upper bounds. For example:

*h*^{−}^{1}^{2}^{−}^{2δ}

*y*2*y*3

*ξ*0*−b*^{1/2}*y*3

*h*^{1/2}

*ϕ*^{2}_{0}(b^{1/2}*h*^{−}^{1/2}*y*3)χ(4h^{−}^{δ}*y*3)χ^{2}

4h^{−}* ^{δ}*(y

_{1}

^{2}+

*y*

_{2}

^{2})

^{1/2}

*dy,*can be estimated by

*O*(h

^{1+δ}). This achieves the proof of Proposition 6.1. ✷

This suggests strongly that the next term in the expansion of*λ(h)*is*O(h*^{4/3}), but to go further
we need to analyze the model more carefully in the neighborhood of the points of the boundary
where*H*(x)*|N(x)*vanishes. For this we need to enter more deeply into the geometry of the
boundary in connection with the magnetic vector field, and this will be done in Section 8.

**7. Rough lower bounds**

We assume that the magnetic field is constant. Let*u** ^{h}* be an eigenfunction associated to the
eigenvalue

*λ(h) = inf Sp(P*

_{A,Ω}*).*

^{h,N}**7.1. A priori estimates**

As proved in [12], we have, without the assumption (1.5), the following behavior of*u** ^{h}*.

PROPOSITION *7.1. – Ift*=*t(x) =d(x, ∂Ω), then, for anyk∈*N*, there exists a constantC**k*

*depending only onk, such that*

*t*^{k/2}*u*^{h}*C**k**h*^{k/4}*u** ^{h}*
(7.1)

*and* *t** ^{k/2}*(hD

*−A)u*

^{h}*C*

*k*

*h*

^{(k+2)/4}

*u*

^{h}*.*

(7.2)

*Proof. – Proposition 6.1 and the fact that*Θ0*<*1give the existence of constants*h*0*>*0 and
*C*0*>*1such that

*C*_{0}^{−}^{1}*bhbh−λ(h)C*0*bh,* *∀h∈*]0, h0].

(7.3)

Of course this is a very rough estimate.

Let us remark that (7.1) and (7.2) are valid when*k*= 0. As for the2-dimensional case in [11],
we proceed by recursion.

By changing*t*away from the boundary, we can assume that*x→t(x) =d(x, ∂Ω)*is extended
as a *C*^{1}(Ω) function. By choosing suitable coordinates and after a gauge transform, we can
assume that

*A*1(x) = 0, *A*2(x) =*−b*

2*x*3*,* *A*3(x) =*b*
2*x*2*.*
As*t/∂Ω = 0, we have by integrating by parts and fork >*0,

*ihb*

Ω

*t*^{k}*|u*^{h}*|*^{2}*dx*=

Ω

*t*^{k}

(hD2*−A*2),(hD3*−A*3)

*u*^{h}*·u*¯^{h}*dx*

=

Ω

*t** ^{k}* (hD3

*−A*3)u

*(hD2*

^{h}*−A*2)u

^{h}*−*(hD2

*−A*2)u

*(hD3*

^{h}*−A*3)u

^{h}*dx*

+*ihk*

Ω

*t*^{k}^{−}^{1}
*∂t*

*∂x*2

(hD3*−A*3)u^{h}*−* *∂t*

*∂x*3

(hD2*−A*2)u^{h}

¯
*u*^{h}*dx*
(7.4)

and

Ω

*t** ^{k}*(hD

*−A)u*

^{h}^{2}

*dx*=

Ω

*λ(h)t*^{k}*|u*^{h}*|*^{2}*−ihkt*^{k}^{−}^{1}

(∇*x**t)·*(hD*−A)u*^{h}

¯
*u*^{h}

*dx.*

(7.5)

If*k*= 1we get from (7.4)
*hb*

Ω

*t|u*^{h}*|*^{2}*dx*

Ω

*t*(hD*−A)u*^{h}^{2}*dx*+*Ch*(hD*−A)u*^{h}*· u*^{h}*,*
(7.6)

and then, we use (7.5) to get
*hb*

Ω

*t|u*^{h}*|*^{2}*dxλ(h)*

Ω

*t|u*^{h}*|*^{2}*dx*+*Ch*(hD*−A)u*^{h}*· u*^{h}*.*
(7.7)

So

*hb−λ(h)*

Ω

*t|u*^{h}*|*^{2}*dxCh*

*λ(h)u*^{h}^{2}*.*
(7.8)

We obtain (7.1) with *k*= 1 from (7.3) and (7.8), and then it is easy to get (7.2) for *k*= 1
from (7.5) by using (7.3), (7.1) with*k*= 1and (7.2) with*k*= 0.

If*k*2,we get in the same way
*hb−λ(h)*

Ω

*t*^{k}*|u*^{h}*|*^{2}*dxhkCt*^{−}^{1+}^{k}^{2}(hD*−A)u*^{h}*· t*^{k/2}*u*^{h}*,*
(7.9)

which gives, using (7.3),

*t*^{k/2}*u*^{h}*Ckt*^{−}^{1+}^{k}^{2}(hD*−A)u*^{h}*.*
(7.10)

Using (7.10) and (7.5) we get

*t*^{k}*|(hD−A)u*^{h}^{2}*Ckh*

*t*^{k/2}*u*^{h}^{2}+*t*^{−}^{1+}^{k}^{2}*|(hD−A)u*^{h}^{2}
*.*
(7.11)

Then we can proceed by recursion. (7.2) for*k*=*j−*2and (7.10) for*k*=*j*give (7.1) for*k*=*j.*

Formulas (7.11) with*k*=*j*and (7.2) for*k*=*j−*2give (7.2) with*k*=*j.* ✷
**7.2. A partition of unity**

Let(χ*γ*(z))*γ**∈Z*^{3}be a partition of unity ofR^{3}*.*For example we can take
*χ**γ**∈C** ^{∞}*(R

^{3};R) and supp(χ

*γ*)

*⊂γ*+ [

*−*1,1]

^{3}

*,*

*∀γ∈*Z

^{3}

*,*

*γ*

*χ*^{2}* _{γ}*(z) = 1 and

*γ*

*χ**γ*(z)^{2}*<∞.*
(7.12)

If*τ(h)*is a function of*h*such that*τ(h)∈*]0, ε(Ω)[, where*ε(Ω)*is the geometric constant which
is the maximal*ε*for the property that*{d(x, ∂Ω)< ε}*is a regular tubular neighborhood inΩ
of*∂Ω, we will define the functions*

*χ**γ,τ(h)*(z) =*χ**γ*

*z/τ(h)*

*,* *∀γ∈*Z^{3}*.*
(7.13)

So we get a new partition of unity such that

*γ*

*χ*^{2}* _{γ,τ(h)}*(z) = 1,

*γ*

*∇χ**γ,τ*(h)(z)^{2}*Cτ*(h)^{−}^{2}*,*
(7.14)

and

supp(χ*γ,τ(h)*)*⊂τ(h)γ*+

*−τ(h), τ(h)*3

*.*
Then, for any*u∈H*^{1}(Ω), we have:

*q*^{h}* _{A}*(u) =

*γ*

*q*_{A}* ^{h}*(χ

*γ,τ(h)*

*u)−h*

^{2}

*|∇χ*

*γ,τ(h)*

*|u*

^{2}

*.*(7.15)

Let us define